$$H_\mathfrak C(A,B)=H_\mathfrak K(A,B).$$
Thus, a full subcategory is completely defined by the class of its objects. Conversely, any subclass of the class of objects of a category $\mathfrak K$ uniquely defines a full subcategory, for which it serves as the class of objects. This subcategory contains exactly those morphisms for which the sources and targets belong to that subclass. In particular, the full subcategory corresponding to a single object $A$ consists of the set $H_\mathfrak K(A,A)$.
|[a1]||B. Mitchell, "Theory of categories" , Acad. Press (1965)|
Full subcategory. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Full_subcategory&oldid=34520